Controllable Production Rate and Quality Improvement in a Two-Echelon Supply Chain Model

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Controllable Production Rate and Quality Improvement in a Two-Echelon Supply Chain Model

Mitali Sarkar, Byung Chung

To cite this version:

Mitali Sarkar, Byung Chung. Controllable Production Rate and Quality Improvement in a Two-

Echelon Supply Chain Model. IFIP International Conference on Advances in Production Management

Systems (APMS), Aug 2018, Seoul, South Korea. pp.238-245, �10.1007/978-3-319-99704-9_29�. �hal-

02164844�

in a two-echelon supply chain model

MITALI SARKAR1[0000-0002-4549-3035] and BYUNG DO CHUNG 2[0000-0003-1878-1338]

1Department of Industrial & Management Engineering, Hanyang University, Ansan, Gyeonggi-do, 15588, South Korea.

2Department of Industrial Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, South Korea.

bd.chung@yonsei.ac.kr

Abstract. The flexible production plays the key role within any production sys- tem. An optimization model is developed for a production system with flexible production rate within a fixed limit, defined by the management, with quality improvement in a supply chain management. The aim of the model is to obtain the best optimum production rate with the global minimum cost. It is assumed that the lead time demand follows a normal distribution and a lead time crashing cost is used to reduce the lead time. A classical optimization technique is used to solve the supply chain model. A theorem is established to obtain the global min- imum total cost. A numerical example is given to illustrate the model. Numerical studies prove that this model converges over the existing literature at the global minimum cost.

Keywords: Flexible Production Rate, Supply Chain Management, Quality Im- provement, Controllable Lead Time.

1 Introduction

Supply chain management (SCM) is introduced by Goyal [1], where he thought about the basic combination of buyer and vendor. There was no especial policy except the adding the basic cost of two players. After a decade, another researcher Banerjee [2]

incorporated a new strategy lot-for-lot (LFL) within SCM model, where it is found that LFL policy is convergent over basic model of Goyal [1]. Just after 2 years of Banerjee’s [2] model, Goyal [3] wrote a short note over a major concept as single-setup-multi- delivery (SSMD) policy. The policy is valid when the buyer’s holding cost is less than vender’s holding cost. But there is a trade-off between holding cost and transportation cost of buyer. Sarkar et al. [4] extended SSMD policy to single-setup-multi-unequal- delivery (SSMUD) policy, where they proved that SSMUD is always convergent over SSMD policy. Even though SSMUD is the best strategy for transportation policy till now, but the trade-off between two different costs exist till now. The whole invention is within the grip of domestic deterministic modelling, where the basic aim of proposed model is to consider a probabilistic modelling with some real extensions.

2

The basic inventory model for SCM with continuous controllable lead time is con- sidered by Ben-Daya and Rauf [5]. The controllable lead time crashing cost with dis- crete lead time crashing is initially utilized by Ouyang et al. [6], whereas that model is modified by Moon and Choi [7]. The concepts of quality improvement and setup cost reduction were introduced by Ouyang et al. [8]. This model of Ouyang et al. [8] was extended by Sarkar and Moon [9] by controllable lead time dependent backorder rate but with continuous investment for setup cost reduction. Sarkar et al. [4] extended the concept of quality improvement with backorder price discounts. Kim and Sarkar [10]

extended the same field with multi-stage production system Majumder et al. [11] de- veloped a SCM model with similar direction for quality improvement. There are several extensions in this direction, but no one considered variable production rate with SSMUD policy in supply chain management (SCM). Nowadays, it is found that con- stant production rate (Glock, [12]) may not support always for different types of rapid production. Glock [13] proved that production rate may be variable or controllable but not constant for all production systems. The production rate should be controllable as if machine moves to out-of-control state, it may produce defective products. Thus, to control defective products or imperfect quality items, production rate should be reduced or due to urgent production, production rate may increase to a certain limit for rapid production. Therefore, production rate is always constant, it is not always valid for all production system. The proposed model considers a SCM model with SSMUD policy under variable production rate, which is the initial stage to make a sustainable supply chain management through sustainable manufacturing which follows a normal distri- bution. The total cost is minimized through a classical optimization technique.

The rest of the paper is organized as follows: Section 2 explains mathematical model of the research problem. Section 3 contains the numerical results of the model. Finally, conclusions and future extensions are discussed in Section 4.

2 Mathematical model

A basic two-echelon supply chain management is considered for single type of products within single-manufacturer and single-buyer. The Production rate 𝑃𝑃 is variable and it is considered as 𝑓𝑓(𝑃𝑃) =𝑎𝑎𝑃𝑃+^𝑏𝑏_𝑝𝑝; a and b are scaling parameters and P is the production rate with production setup cost 𝑆𝑆 per setup. The unit production cost is production de- pendent (see for reference Sarkar et al. [14]) and the quality improvement with contin- uous investment is considered here (see for reference Kim and Sarkar [10]). If the buyer orders Q items, manufacturer produces zQ items and transports them into z times of shipments to buyer. The reorder point R of each buyer is allowed when the inventory touches the reorder point 𝑅𝑅=𝐷𝐷𝐷𝐷+𝑘𝑘σ√𝐷𝐷, where 𝐷𝐷 is the demand of products during unit time, 𝐷𝐷 is average lead time, DL is the expected demand during lead time, kσ√𝐷𝐷 is safety stock, and k is safety factor. Safety factor is considered as a decision variable instead of reorder point. Let us suppose 𝐷𝐷0≡ ∑𝑛𝑛 𝑣𝑣𝑗𝑗

𝑗𝑗=1 and 𝐷𝐷𝑖𝑖 be the length of lead time with components 1, 2,....,i crashed to their minimum duration. Thus, 𝐷𝐷_𝑖𝑖 can be ex- pressed as 𝐷𝐷_𝑖𝑖≡ ∑^𝑖𝑖_𝑗𝑗=1(𝑣𝑣_𝑗𝑗−𝑢𝑢_𝑗𝑗),𝑖𝑖= 1,2 … . ,𝑛𝑛; where 𝑢𝑢_𝑗𝑗 and 𝑣𝑣_𝑗𝑗 are the minimum dura- tion and normal duration to crash the lead time. The lead time crashing cost per cycle C(L) is expressed as 𝐶𝐶(𝐷𝐷) = 𝑐𝑐_𝑖𝑖(𝐷𝐷_𝑖𝑖−1− 𝐷𝐷) +∑_𝑗𝑗=1^𝑖𝑖−1𝑐𝑐_𝑗𝑗�𝑣𝑣_𝑗𝑗− 𝑢𝑢_𝑗𝑗�,𝐷𝐷 ∈[𝐷𝐷_𝑖𝑖,𝐷𝐷_𝑖𝑖−1], where 𝑐𝑐_𝑖𝑖

is the crashing cost per unit time. Due to controllable lead time shortages are allowed and fully backlogged.

The average total cost of manufacturer is TC_M(z, Q, P,γ) =SD

zQ + h^m zQ

2 �z�1− D

f(P)� −1 + 2D

f(P)�+yDzQγ

2 +�Cm+ Cd

f(P) +αf(P)�D + B ln�γ0

γ � +F1D

Q . (1)

For the manufacturer, the average total cost is combination of setup cost, holding cost, investment for imperfect products, variable production cost, quality improvement cost, and transportation cost, respectively. Here ℎ_𝑚𝑚 is the holding cost of the manufac- turer per unit product per unit time, 𝑦𝑦 is the investment for quality improvement, 𝐶𝐶𝑚𝑚

and 𝐶𝐶_𝑑𝑑 are material cost and development cost, 𝛼𝛼 is the tool/dye cost, 𝛾𝛾 is the probabil- ity for shifting “in-control’’ state to “out-of-control’’ state of the manufacturing process, 𝛾𝛾₀ is the initial probability to go “out-of-control’’ state, B is the scaling parameter re- lated with the quality of product, and 𝐹𝐹1 is the transportation cost of the manufacturer.

The average total cost of buyer is 𝑇𝑇𝐶𝐶𝐵𝐵(𝑄𝑄,𝑘𝑘,𝐷𝐷) =�𝐴𝐴𝐷𝐷

𝑄𝑄 +𝐷𝐷𝐶𝐶(𝐷𝐷)

𝑄𝑄 +ℎ𝑏𝑏�𝑄𝑄

2 +𝑅𝑅 − 𝐷𝐷𝐷𝐷�+𝜋𝜋𝐷𝐷

𝑄𝑄 𝐸𝐸(𝑋𝑋 − 𝑅𝑅)⁺�, (2) which is the summation of ordering cost, lead time crashing cost, holding cost, and shortage cost. Here 𝐴𝐴 is the ordering cost per order and ℎ𝑏𝑏 is the holding cost of the buyer per unit per unit time, and 𝜋𝜋 is shortage cost per unit shortage.

The expected shortage at the end of the cycle can be written as 𝐸𝐸(𝑋𝑋 − 𝑅𝑅)⁺=�^∞(𝑥𝑥 − 𝑅𝑅)𝑑𝑑𝐹𝐹(𝑥𝑥)

𝑅𝑅

=𝜎𝜎√𝐷𝐷 ψ(𝐾𝐾) (3) ,where 𝜓𝜓(𝑘𝑘) =𝜙𝜙(𝑘𝑘)− 𝑘𝑘[1− Φ(𝑘𝑘)], 𝜙𝜙 stands for the standard normal probability density function and Φ stands for the cumulative distribution function of normal distri- bution.

Total supply chain cost can be found as the summation of total cost of two players.

Therefore,

𝑇𝑇𝐶𝐶𝑆𝑆(𝑧𝑧,𝑄𝑄,𝑃𝑃,𝛾𝛾,𝑘𝑘,𝐷𝐷) = ^SD_zQ+ hmzQ

2 �z�1−_f(P)^D � −1 +_f(P)^2D�+^yDzQγ₂ +

�Cm+_f(P)^Cd +αf(P)�D + B ln�^γ_γ⁰�+^F_Q^1D+�^{𝐴𝐴𝐴𝐴}_𝑄𝑄 +^{𝐴𝐴𝐷𝐷(𝐿𝐿)}_𝑄𝑄 +ℎ𝑏𝑏�^𝑄𝑄₂+𝑅𝑅 − 𝐷𝐷𝐷𝐷�+ ^{𝜋𝜋𝐴𝐴}_𝑄𝑄 𝐸𝐸(𝑋𝑋 − 𝑅𝑅)⁺�

(4) Taking partial differentiation with respect to 𝑄𝑄, and after simplification, one can be obtained

𝑄𝑄=�^{𝐴𝐴𝐴𝐴+𝐹𝐹1}𝐴𝐴+𝐴𝐴𝐷𝐷(𝐿𝐿)+𝜋𝜋𝐴𝐴𝜋𝜋√𝐿𝐿ψ(𝐾𝐾)+^{𝑆𝑆𝑆𝑆}_𝑧𝑧 ℎ𝑚𝑚𝑧𝑧

2 �𝑧𝑧�1−_{𝑓𝑓(𝑃𝑃)}^𝑆𝑆 �−1+_{𝑓𝑓(𝑃𝑃)}^2𝑆𝑆�+^{𝑦𝑦𝑆𝑆𝑧𝑧𝑦𝑦}₂ +^ℎ𝑏𝑏₂ (5)

4

Partially differentiating equation (4), with respect to 𝑃𝑃 and after simplifying, one can find

𝑃𝑃=^�

1𝛼𝛼�𝐷𝐷_𝑑𝑑−^{ℎ𝑚𝑚𝑧𝑧𝑧𝑧}₂^{(𝑧𝑧−2)}�+�¹_𝛼𝛼�𝐷𝐷_𝑑𝑑−^{ℎ𝑚𝑚𝑧𝑧𝑧𝑧}₂^{(𝑧𝑧−2)}�−4𝑎𝑎𝑏𝑏

2𝑎𝑎 (6)

For obtaining the optimum value of 𝑘𝑘, one can take the partial differentiation of the total supply chain cost with respect to 𝑘𝑘 and after simplification, one can have

ℎ𝑏𝑏𝜎𝜎√𝐷𝐷+^{𝜋𝜋𝐴𝐴𝜋𝜋√𝐿𝐿}_𝑄𝑄 (Φ(𝑘𝑘)−1) = 0 . (7) Solving the equation, it can be found

Φ(𝑘𝑘) = 1−^ℎ_{𝐴𝐴𝜋𝜋}^{𝑏𝑏𝑄𝑄} . (8) Differentiating partially twice with respect to 𝐷𝐷 of the total cost function, it can be ob- tained

^{𝜕𝜕2𝑇𝑇𝐷𝐷𝑆𝑆}

𝜕𝜕𝐿𝐿² =−¹₄�^ℎ_𝐿𝐿^𝑏𝑏_{3 2}^{𝑘𝑘𝜋𝜋}⁄ +𝜋𝜋𝐴𝐴𝜋𝜋ψ(𝐾𝐾)

𝑄𝑄𝐿𝐿^{3 2⁄} � (9)

The total cost function 𝑇𝑇𝐶𝐶𝑆𝑆 is concave with respect to L as second order derivative with respect to L is less than zero. For optimum value of 𝛾𝛾, differentiating the total cost function and simplifying, one can obtain,

γ=_{𝑌𝑌𝐴𝐴𝑧𝑧𝑄𝑄}^2𝐵𝐵 . (10) These values of the decision variables are optimum values as the principal minors are positive definite always.

3 Numerical experiment

This section consists of experiment’s result and analysis from the results.

3.1 Numerical example

To test the model, an illustrative numerical experiment is conducted. The experiment is considered with a tool as MATLAB 17B. For the experiment, the input parameters’

value are taken from Majumder et al. [11] and given below:

The demand of products is considered here 600 unit/cycle. The ordering cost of the retailer is $200/order. Shortages are allowed in this model and it is considered as

$10/unit shortage. Safety σ has the fixed value of 7. Setup cost is assumed as

$200/setup. γ0 is the initial probability to shift “out-of-control” state 0.0002;

𝐹𝐹₁=$.1/shipment; The investment for quality improvement is $175. The holding cost of the manufacturer is considered as $0.13/unit/unit time and buyer as $0.18/unit/unit time.

Material cost, development cost and tool/dye cost are assumed as $3/setup, $10/setup, and $0.001/cycle, respectively. Scaling parameters’ values are 𝑎𝑎=0.05 and 𝑏𝑏=500, and 𝐵𝐵=400. Lead time crashing cost is $22.4.

The numerical experiment has been done with the given values of the parameters.

The model obtains the global optimal solution using classical optimization technique in MATLAB coding. The optimum values of equations (5) ~ (10) are used to obtain the optimum results. The optimal resultsare obtained as follows: the optimum supply chain cost 𝑇𝑇𝐶𝐶_𝑆𝑆=$5929.88; the optimal lot size (𝑄𝑄^∗)= 28.26unit; the optimal probability of

shifting to “out-of-control” state from “in-control” state 𝛾𝛾^∗= 0.00009; the optimal pro- duction rate (𝑃𝑃^∗)= 3.33units; 𝑘𝑘^∗= 1.37the number of shipments (𝑧𝑧)=3; lead time (𝐷𝐷)=4. It is found that optimum production rate is more than the demand, which indi- cates that there should not be any shortages any time and the model is validated with the theoretical perspective based on the taken assumptions. Based on the probability value of shifting to “out-of-control” state from “in-control” state, it can be concluded that the quality of products is improved as the probability is reduced, and finally the shipment number indicates that the model follows single-setup-multi-delivery policy.

If we compare this research model with Majumder et al. [11], then the cost of their model is $ 6393.17 under same assumptions whereas the proposed model gives the cost

$5929.88. Therefore, the savings from the existing literature is 7.18%.

3.2 Sensitivity analysis

A sensitivity analysis is performed here to estimate the variation of total cost of supply chain with change of key parameters of the model. The optimal value of total cost is changed (-10%, -5%, +5%, +10%) with the change of different parameters are given below in Table 1.

Table 1. Sensitivity analysis table Param-

eters

Changes (in %)

*

TCs

(in %)

Param- eters

Changes (in %)

*

TCs

(in %)

A -10 -0.05 π -10 -0.004

-5 -0.03 -5 -0.001

+5 +0.02 +5 -0.0001

+10 +0.05 +10 +0:001

S -10 -0.02 σ -10 +0.03

-5 -0.01 -5 +0.01

+5 +0.01 +5 -0.01

+10 +0.02 +10 -0.03

γ

0 ^-10 _-5 ^-0.007 _-0.003 ^{F1 -10} _-5 ^-0.00003 _-0.00001

+5 +0.003 +5 +0.00001

+10 +0.006 +10 +0.00003

Y -10 -0.007 α -10 -0.002

-5 -0.003 -5 -0.001

+5 +0.003 +5 +0.001

+10 +0.006 +10 +0.002

h

m -10 -0.01 ℎ𝑏𝑏 -10 -0.03

-5 -0.006 -5 -0.01

+5 +0.005 +5 +0.01

+10 +0.01 +10 +0.03

Cm -10 -0.03 -10 +0.001

6

-5 -0.02

C

d ^{-5 +0.0005}

+5 +0.02 +5 -0.0003

+10 +0.03 +10 -0.0005

a -10 +0.002 b -10 -0.002

-5 +0.0008 -5 -0.0008

+5 -0.0007 +5 +0.0008

+10 -0.001 +10 +0.002

B -10 -0.02

-5 -0.008

+5 +0.008

+10 +0.01

From Table 1, one can say that ordering cost, setup cost, material cost, backlogging cost, and transportation cost are changing as they are directly proportional to the total cost of the supply chain. Scaling parameters (a, b) of flexible production rate are in- versely and directly proportional, respectively to the total cost. For the controllable pro- duction rate development cost is inversely proportional to total cost of the supply chain.

4 Conclusions

Two-echelon supply chain model was considered where single-setup-multiple-delivery was applied. Production rate was variable and controllable. Manufacturing system al- ways maintained quality of products. Buyer faced some shortages, but it was fully back- logged. To continue the business supply chain players had to follow some strategies.

The model had been solved analytically and numerically. Analytically quasi-closed form solutions were obtained and from numerical study optimal total cost were gained.

This model can be easily applied in any production system, where the production sys- tem is flexible and producing single-type of products. The model can be used where the production system is passing through a long-run production system and there is a prob- ability for the production system moves from “in-control’’ state to “out-of–control’’

state; the policy from this model can be employed to improve the quality of products.

This model can be extended with the setup time and transportation time-dependent lead time demand, which may be considered as controllable. One can invest some invest- ments to reduce the setup time or transportation time or both. It may be considered that second setup is dependent on first setup and hence the lead time is dependent only setup time, which may be controllable also.

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